A fifth dimension for "historical time". :)
So, then all four dimensions of space-time are flowing in time!
Two places where this played out, in science fiction, were the Premonition episode of The Outer Limits, where a husband and wife were caught up in a kind of side-ways time stream along a fifth dimension that was nearly orthogonal to the flow of real time; and in Steven King's Langoliers. Both stories made tacit reference to the idea of real versus imaginary time, in which in the latter dimension, oscillatory processes became exponentially decaying. So it was difficult to start and maintain fires (in Premonition) and jet fuel lost much of its effectiveness (in Langoliers).
Since you would want the time flow to be some kind of universal reference, it should be invariant. Additionally, if it matches each of our experience of time flow, then it should match the proper time for the space-time geometry. So, the natural choice is to just add in the proper time ($s$) alongside the spatial coordinates ($x,y,z$) and real time coordinate ($t$) of spacetime.
In the Minkowski geometry of Special Relativity,
$$dx^2 + dy^2 + dz^2 - c^2 dt^2$$
is an invariant which, when negative, equates to $-c^2 ds^2$, giving you the differential relation for proper time. So, if we throw in the coordinate then we can rewrite the line element as a constraint:
$$dx^2 + dy^2 + dz^2 - c^2 dt^2 + c^2 ds^2 = 0.$$
So, now the time dilation itself can be expressed in terms of the coordinates: $s - t$. Notably, this continues to have meaning even in the non-relativistic limit $c → ∞$, provided we scale it up as $u ≡ c²(s - t)$.
The line element, expressed in terms of $t, u$ becomes
$$dx^2 + dy^2 + dz^2 + 2 dt du + {du^2 \over c^2} = 0,$$
and the invariant $ds$ is now replaced by $dt + {du^2 \over c^2}$. In the non-relativistic limit, these become the constraint
$$dx^2 + dy^2 + dz^2 + 2 dt du = 0,$$
and invariant $dt$, respectively.
For the non-relativistic world, real time and historical time become one and the same; while a vestige of time dilation lives on as $u$. In the non-relativistic limit, the $u$ coordinate for a body moving with velocity $𝐫̇(t) = 𝐯(t)$ can be expressed as
$$u = u_0 - \int_0^t {1 \over 2} v^2 dt$$
which is also (up to an additive constant) the negative of the action per unit mass for the body.
The resulting geometry is known as the Bargmann Geometry. The symmetries which leave both the Bargmann metric and the linear $dt$ invariant fixed is the Bargmann Group, which is the central extension of the Galilei Group and is the symmetry group more correctly suited for the non-relativistic world. For instance, the mass $m$, momentum $𝐩 = (p_x,p_y,p_z)$ and kinetic energy $H$ of a body transform under a change of velocity $𝐯$ as a Bargmann 5-vector
$$(m,𝐩,H) → (m, 𝐩 ‒ m𝐯, H - 𝐩·𝐯 + ½mv²),$$
and combine with the coordinates to produce an invariant 1-form
$$m du + p_x dx + p_y dy + p_z dz - H dt.$$
We need only expand Hamilton's equations to handle the more general case where $H$ not interpreted as the Hamiltonian and also includes the potential energy,
$${\partial H \over \partial m} = {du \over dt}, {\partial H \over \partial 𝐩} = 𝐯 = {d𝐫 \over dt},$$
$$-{\partial H \over \partial u} = {dm \over dt}, -{\partial H \over \partial 𝐫} = 𝐅 = {d𝐩 \over dt}.$$
For instance, in the the presence of a conservative force satisfying the equivalence principle, given by a potential per unit mass $U$ (e.g. $U = -{GM \over r}$ for a gravitating body of mass M), we can write the Hamiltonian as
$$H = {p^2 \over 2m} + mU,$$
and convert it to a quadratic constraint
$$p^2 - 2mH + 2m^2 U = 0.$$
Applying Hamilton's equations, we get
$$2𝐩 - 2m {d𝐫 \over dt} = 𝟎, -2H - 2m{du \over dt} + 4mU = 0,$$
$$2m{d𝐩 \over dt} + 2m^2 {\partial U \over \partial 𝐫} = 𝟎, -2m{dm \over dt} = 0,$$
from which we obtain (with the aid of the quadratic constraint)
$$𝐩 = m {d𝐫 \over dt}, H = m\left({1 \over 2} v^2 + U\right),$$
$$\left|{d𝐫 \over dt}\right|^2 + 2 {du \over dt} - 2U = 0.$$
If we multiply the last of these equations by $dt^2$, we obtain the following metric constraint
$$dx^2 + dy^2 + dz^2 + 2 dt du - 2U dt^2 = 0,$$
which may be identified as the 5D curved space-time metric for the force associated with the potential $U$ per unit mass. In fact, the geodesics for this metric produce the orbits arising from $U$.
In the Relativistic case, the corresponding geometry does not have any name that I am aware of. Both the Bargmann line element and its relativistic version are metrics for a 4+1 dimensional geometry. In the non-relativistic case, both $s$ and $t$ are null directions with respect to the metric, while in the relativistic case, $t$ is time-line, while $s$ is space-like. Thus, the relativistic and non-relativistic versions have the same 4+1 dimensional geometry in common, but differ in that the "proper time" coordinate $s$ is null in the non-relativistic world, but spacelike in the relativistic world; the same applies to the coordinate $u$.
The relativistic version of the Newtonian gravity metric is equivalent to the Schwarzschild metric. The line element for both can be written as members of a 1-parameter family of metrics:
$$dx^2 + dy^2 + dz^2 + 2 dt du - {\alpha du^2} - 2U dt^2 - {2 \alpha U \over 1 + 2 \alpha U} dr^2 = 0,$$
with the proper time invariant $ds = dt + \alpha du$, the potential per unit mass $U = -{GM \over r}$ and $r = \sqrt{x^2 + y^2 + z^2}$. For Newtonian gravity, $\alpha = 0$, while for the Schwarzschild metric, $\alpha = {1 \over c^2}$.
So, one way of putting all of this together is to think of the 5th dimension as the arena for alternate time-lines. At different layers are different versions of the 4-dimensional spacetime, which we think of as flowing in the $s$ direction - a direction for absolute time or historical time, while the $t$ coordinate describes the coordinate time, or real time ... all of which is just like Lorentz conceived of it, before Einstein came along. A time flow is embedded into the 4-dimensional manifold by associating a "now" at each $s$ on the 4-dimensional manifold. So, the combined geometry describes a moving "now", rolling up the 4-dimensional continuum, as we move from layer to layer up in the $s$ direction.
For Relativity, the canonical 1-form can be written in different ways, depending on which two of the three coordinates $(s,t,u)$ are chosen:
$$p_x dx + p_y dy + p_z dz - H dt + m du,$$
$$p_x dx + p_y dy + p_z dz - H ds + M du,$$
$$p_x dx + p_y dy + p_z dz - E dt + mc^2 ds,$$
where $M = m + {H \over c^2}$ is the "moving mass" and $E = Mc^2 = mc^2 + H$ is the "total energy". The mass shell invariant can be written as
$$p^2 - 2MH + {H^2 \over c^2} = p^2 - {E^2 \over c^2} + m^2 c^2 = 0,$$
and is supplemented by the linear invariant $m = M - {H \over c^2}$.
This geometry, strictly speaking, goes beyond Relativity since the symmetry group associated with it not only includes the Poincaré group, but an extra generator corrsponding to translation for the $u$ coordinate. Its representations differ from those of Relativity in that they also allow for (1) non-zero values for the quadratic invariant, (2) linear invariants that differ from the mass $m$ and (3) non-zero rest frame values for $H$ (more generally: they make the zero point for $H$ relative, as it is in the non-relativistic world). If $H$ has a rest-frame value of $U ≠ 0$, then the two invariants will become:
$$p^2 - 2MH + {H^2 \over c^2} = -2mU + {U^2 \over c^2},$$
$$M - {H \over c^2} = m - {U \over c^2}.$$
Another notable difference from the representations in Relativity is that while the vacuum is characterized by $(E,𝐩) = (0,𝟬)$, in the expanded geometry, this condition becomes $(M,𝐩) = (0,𝟬)$, while $H = U ≠ 0$ is allowed. There is extra room provided for a notion of vacuum energy, which is not accounted for properly or at all, in the representations of the Poincaré group.
Another interesting feature of the flat space metric is that when it is written in terms of $s$ and $t$, rather than $t$ (or $s$) and $u$
$$\Re\left(dx^2 + dy^2 + dz^2 - c^2 (dt + i ds)^2\right)$$
it becomes a metric for an expansion of Minkowski geometry with complex time $t + is$. So, this may also provide the basis for a geometrical interpretation of complex time.
